10
H. Bichsel and H. Schindler
Fig. 2.3 Dielectric loss
function Im (−1/ε (E)) of
solid silicon [31] as a function
of the photon energy E
1
−
10
1
10
2
10
3
10
4
10
E [eV]
7
−
10
6
−
10
5
−
10
4
−
10
3
−
10
2
−
10
1
−
10
1
10
Im(-1/ε)
gap
band
peak
plasmon
edge
23
L
edge
K
conservation of energy and momentum, the photon energy E after the collision and
the scattering angle θ of the photon are then related by
E
=
mc 2
1 − cos θ + (1/u)
,
(2.9)
where u = E/
mc 2
is the photon energy (before the collision) in units of the
electron rest energy.
The kinetic energy T = E − E imparted to the electron is largest for a headon collision (θ = π) and the energy spectrum of the recoil electrons consequently
exhibits a cut-off (Compton edge) at
T max = E
2u
1 + 2u
.
The total cross section (per electron) for the Compton scattering of an unpolarised photon by a free electron at rest, derived by Klein and Nishina in 1929 [33],
is given by
σ (KN) = 2π
α ¯
hc
mc 2
2 1 + u
u 2
2 (1 + u)
1 + 2u
−
ln (1 + 2u)
u
+
ln (1 + 2u)
2u
−
1 + 3u
(1 + 2u) 2
.
(2.10)
At low energies (u 1), the Klein-Nishina formula (2.10) is conveniently
approximated by the expansion [34]
σ
(KN)
=
8π
3
α ¯
hc
mc 2
2
Thomson cross section
1
(1 + 2u)
2
1 + 2u +
6
5
u
2
+ . . .
,